Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh

Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

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Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh Amine Benhfid, El Bachir Ameur ⇑, Youssef Taouil Research Team MSISI – LaRIT, Department of Computer Science, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco

a r t i c l e

i n f o

Article history: Received 7 May 2018 Revised 20 August 2018 Accepted 17 September 2018 Available online xxxx MSC 2018: 03C40 68P20 94A08 00A05 68P30

a b s t r a c t Data hiding is the discipline of hiding information into digital media. A reversible steganographic method allows to extract data from the stego file as well as to retrieve the cover file. This paper proposes a reversible steganographic method based on interpolation by linear box-splines on the three directional mesh. Secret data is hidden in the error between the cover and interpolated pixels. To have better imperceptibility, we search for the nearest value to the cover pixel that can dissimulate data; this was achieved with two techniques: Optimal Pixel Adjustment Procedure and Message Adaptive Error. The proposed method is tested through experiments on a variety of images and compared to prior works. Results indicate a large capacity of hiding and good values of PSNR. The proposed work was also tested by steganalysis attacks; results show a low detectability rate. We find in the comparison that the proposed work surpasses the literature in the capacity of hiding with an equivalent level of imperceptibility. Ó 2018 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Steganography is a data hiding technique, it consists of the dissimulation of a secret information into digital files so that an intended recipient can extract it successfully. These files can be texts, images, audio or videos; However, the most utilized one is image since it is shared widely everyday on the networks services; and most importantly because of the insensibility of the human visual system to small changes within images. The main objective of a steganographic system is that no one shall suspect the existence of the hidden information. The dissimulation of a secret data produces a new image which is called the stego image. To achieve the objective of steganography, it is primordial to keep as low as possible the distortions brought upon the cover image by the dissimulation process. The steganographic system is evaluated by the imperceptibility, the capacity and the security. The imperceptibility is reached when the human eye can not distinguish between the cover and the ⇑ Corresponding author. E-mail address: [email protected] (E.B. Ameur). Peer review under responsibility of King Saud University.

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stego images. The capacity is the maximum number of bits of the secret data that can be hidden in the cover image without causing perceptible artifacts. The security refers to the undetectability of the secret data inside the stego image. The most common and well-known steganographic method is called least significant bit (LSB) substitution; it embeds secret data by substituting the LSB of a pixel by the secret bit (Bender et al., 1996). Many optimized LSB methods have been proposed later to break the uniformity of the histogram in this technique (Wang et al., 2001; Chan and Chen, 2004; Lin et al., 2009). It is a good steganographic mechanism since the changes in a least significant bits yield few changes in the original image. The human perceptibility is sensitive to big changes in the pixels of the smooth areas; while it is not sensitive to changes in the edge areas. In Fridrich et al. (2001), authors describe a very accurate and reliable method that can detect LSB embedding in randomly scattered pixels in both 24-bit color images and 8-bit gray-scale or color images. Not all pixels in a cover image can tolerate equal amount of changes without causing noticeable distortion. Hence, to improve the quality of the stego images, several adaptive methods have been proposed in which the amount of bits to be embedded in each pixel is variable (Wu and Tsai, 2003; Wu et al., 2005; Liao et al., 2011). In reversible steganography, both the secret information and the cover image are retrieved by the recipient; this condition is required in some applications of steganography. In Jung and Yoo

https://doi.org/10.1016/j.jksuci.2018.09.016 1319-1578/Ó 2018 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

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A. Benhfid et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

(2009), a reversible data hiding scheme based on interpolation is proposed. The scaling up allows to find the pixels of interpolation; those pixels are used to construct the interpolated pixels. Then, data is hidden in the error between the initial and the interpolated pixels. The obtained capacity is good, but it could be improved. In Wang et al. (2013), the algorithm utilized the same methodology while the hiding is based on the histogram shifting. The maximum capacity is not very large for interpolation based steganography; and this lack in the capacity is not compensated in PSNR. The steganographic scheme proposed in Hu and Li (2015) is based on extended interpolation; the capacity is enlarged by maximizing the error between neighboring pixels. The obtained capacity and imperceptibility are good but require improvement. Authors in Lee and Huang (2012) designed an efficient information hiding scheme based on interpolation by neighboring pixels (INP) in order to increase the payload. In Tang et al. (2014), the proposed highCapacity Reversible Steganography (CRS) improved the capacity of hiding obtained by the INP scheme. In Ou et al. (2012), authors proposed a reversible watermarking based on optional predictionerror histogram modification to improve the watermarked image fidelity (or quality) at high embedding rate. In Pei et al. (2013), the proposed algorithm improved the embedding capacity by using histogram shifting and adaptive embedding. In Hu and Li (2015), a high capacity image steganographic scheme based on an extended interpolation method is proposed. In the premise of image quality assurance, the proposed scheme increases the capacity by maximizing the difference between neighbouring pixels. In this paper, a reversible steganographic scheme based on interpolation by bivariate box-splines on the three directional mesh is proposed. To enlarge the capacity without degrading the imperceptibility, data is dissimulated into the difference between the interpolated pixel and either the maximum or the minimum of the four interpolation reference pixels. We choose between the minimum and maximum pixels the one that gives the greater difference with the interpolated pixel. Firstly, the secret data is added to the interpolated pixels as an error;which is adapted in order to have stego pixels as near as possible to the cover pixels. Secondly, data is hidden by substituting the LSBs of the interpolated pixels; then the Optimal Pixel Adjustment Procedure (OPAP) is utilized to minimize the difference between the obtained pixel and the cover pixel. To test the performance of the proposed work, experiments were accomplished on images with different degrees of complexity; and comparison to existing works was carried out. The proposed work provides very large capacity with a good compromise to PSNR. The proposed work is also tested by steganalysis attacks; results indicate a very low rate of detectability. Compared

to literature, results show that the proposed scheme has large capacity of hiding while keeping the PSNR at a good level. The remaining of the paper is organized as follows: Section 2 presents the utilized interpolation methods. In Section 3, the proposed scheme is explained in details. Section 4 discusses experimental results of tests and comparison to literature. Section 5 concludes the paper. 2. Related works 2.1. Neighbour mean interpolation (NMI) Jung et al. proposed in Jung and Yoo (2009) the neighbour mean interpolation (NMI) hiding method. This scheme enlarges an original image to generate a predicted image and then embeds secret information in the predicted image to obtain the stego-image. The receiver reduces the stego image and extracts the secret information from it. Jung et al.’s scheme is shown schematically in Fig. 1; Fig. 1 (a) shows the original image O. A predicted pixel I is inserted between any two horizontal or vertical pixels. Fig. 1 (b) depicts the pixel reference C and Fig. 1 (c) shows the interpolated image. The predicted pixel I is computed as follows:

Iði;jÞ ¼

k 8 jðO ði;j1Þ þOði;jþ1Þ Þ > > 2 > > > > >
þO

Þ

if i ¼ 2m; j ¼ 2n þ 1

ði1;jÞ ðiþ1;jÞ if i ¼ 2m þ 1; j ¼ 2n 2 > > > > > > > jðOði1;j1Þ þOði1;jÞ þOði;j1Þ Þk : otherwise; 3

ð1Þ

where m and n are the height and width of the cover image C, and (i; j) denotes the pixel’s position. The secret message is then concealed into the predicted pixel to generate the stego pixel. The receiver reduces the stego image to recover the original image and extracts the secret message from the stego-image. In Fig. 1 (a), the reference pixels of the original image are Oð0;0Þ ¼ 145; Oð0;1Þ ¼ 56; Oð1;0Þ ¼ 116 and Oð1;1Þ ¼ 101. Suppose that the secret message is msg ¼ ð00111110100000Þ2 . We calculate the interpolation pixels Ið0;1Þ ; Ið1;0Þ and Ið1;1Þ using the Eq. (1). We find that Ið0;1Þ ¼ 145þ49 ¼ 97 and Ið1;0Þ ¼ 145þ77 ¼ 111 respec2 2 tively. The virtual predictive value of the intermediate pixel Ið1;1Þ is the average of C ð0;0Þ , Ið0;1Þ and Ið1;0Þ , we find that Ið1;1Þ ¼ ð145 þ 97 þ 111Þ=3 ¼ 118. Next, the differences E1 ; E2 and E3 are calculated by subtracting the references pixel C ð0;0Þ from the three virtual predicted values Ið0;1Þ ; Ið1;0Þ and Ið1;1Þ . The equation is as follows:

Fig. 1. Steps of NMI method using numerical values.

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

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8 > < E1 ¼ jIð0;1Þ  C ð0;0Þ j E2 ¼ jIð1;0Þ  C ð0;0Þ j > : E3 ¼ jIð1;1Þ  C ð0;0Þ j:

  3  C min þ C max AD ¼ 4

Following the same example above, the differences are E1 ¼ 145  97 ¼ 48; E2 ¼ 145  111 ¼ 34 and E3 ¼ 145  118 ¼ 27. By applying log2 on these differences, we find the decimal value as follows: L1 ¼ ½log2 ð48Þ ¼ 5; L2 ¼ ½log2 ð34Þ ¼ 5 and Li L3 ¼ ½log2 ð27Þ ¼ 4. Each Li denotes how much bits of the secret message will be hidden in its corresponding interpolated pixel. Now, we create the stego image using the interpolation image and the secret message msg. Since L1 ¼ 5, we embed 5 bits of the secret message msg ¼ ð00111Þ2 ¼ 7 into Ið0;1Þ ¼ 97. Thus, Sð0;1Þ ¼ 97 þ 7 ¼ 104. Using the same method we obtain the other pixels of stego image.

The predicted value Iði;jÞ is calculated using the neighboring pixels and the value AD defined in Eq. (3):

Iði;jÞ ¼

k 8 jADþðO ði;j1Þ þOði;jþ1Þ Þ=2 > > 2 > > > > >
Iði;jÞ

Þ=2

ð4Þ

The differences Ek ði; jÞ are computed by the comparison between the virtual predicted value I and the mean value of C max and C min . The formula is given as follows:

( E1 ði; jÞ ¼

k 8 jO ði;j1Þ þðOði;j1Þ þOði;jþ1Þ Þ=2 > if i ¼ 2m; j ¼ 2n þ 1 > 2 > > > > > > > > > j k > > : Iði1;jÞ þIði;j1Þ otherwise:

þO

if i ¼ 2m; j ¼ 2n þ 1

ði1;jÞ ðiþ1;jÞ if i ¼ 2m þ 1; j ¼ 2n 2 > > > > > j k > > Oði1;j1Þ þIði1;jÞ þIði;j1Þ : otherwise: 3

2.2. Interpolation by neighboring pixels (INP) In Lee and Huang (2012) proposed a hiding scheme based on interpolation by neighbouring pixels (INP). In their scheme, the virtual predicted value I is the average of the reference pixel C ð0;0Þ and average of two adjacent pixels as the virtual predicted value. The formula is as follows:

ð3Þ

( E2 ði; jÞ ¼ ( E3 ði; jÞ ¼

C max  Iði;jþ1Þ

max ifIði;jþ1Þ < C min þC 2

Iði;jþ1Þ  C min

max ifIði;jþ1Þ P C min þC ; 2

C max  Iðiþ1;jÞ

max ifIðiþ1;jÞ < C min þC 2

Iðiþ1;jÞ  C min

max ifIðiþ1;jÞ P C min þC ; 2

C max  Iðiþ1;jþ1Þ

max ifIðiþ1;jþ1Þ < C min þC 2

Iðiþ1;jþ1Þ  C min

max ifIðiþ1;jþ1Þ P C min þC : 2

ð5Þ

ð2Þ

2

The example used in this section is the same as the example in Section 2.1. Using the Eq. (2), we obtain Ið0;1Þ ¼ 145þð145þ49Þ=2 ¼ 121 2 ¼ 128. The virtual predictive value of the and Ið1;0Þ ¼ 145þð145þ77Þ=2 2 intermediate pixel is the average of Ið0;1Þ and Ið1;0Þ , we get Ið1;1Þ ¼ 121þ128 ¼ 125. Next, the difference is computed to calculate 2 the length of the secret message that can be concealed into the pixel. The difference is the distance between the maximum value and the predicted value. The maximum value is the largest one in the block of four original pixels, it is computed as follows:

Max ¼ maxfC ð0;0Þ ; C ð0;1Þ ; C ð1;0Þ ; C ð1;1Þ g: The differences Ei are obtained by subtracting the predicted values from this maximum value.

8 > < E1 ¼ jMax  Ið0;1Þ j E2 ¼ jMax  Ið1;0Þ j > : E3 ¼ jMax  Ið1;1Þ j: For example, we consider Max ¼ maxf145; 49; 77; 46g ¼ 145; we obtain E1 ¼ 24; E2 ¼ 17 and E3 ¼ 20. 2.3. High-capacity reversible steganography (CRS)

3. Proposed scheme To control the image quality, the proposed scheme considers the complexity of the reduced pixels to adjust the length of the secret message. High-complexity areas are associated with long secret messages, whereas smooth areas are associated with short secret messages. This variance is used to judge whether an area is smooth or complex. In this section, we introduce the interpolation method by bivariate linear box-spline on three directional mesh, which guaranties that the interpolation error E has an approximation order of 2

Oðh Þ where h is the mesh step. Based on this interpolation method, we propose a reversible steganographic scheme that hides the secret data in the error between the cover and the interpolated pixels. In order to improve the imperceptibility by finding the nearest value to the cover pixel that can dissimulate data, we use Optimal Pixel Adjustment Procedure (OPAP) and Message Adaptive Error (MAE) methods. 3.1. Interpolation by linear box spline on three directional mesh Let D be the three directional mesh which is the uniform triangulation of the plane whose set of vertices is Z2 and whose edges are parallel to the three directions e1 ¼ ð1; 0Þ; e2 ¼ ð0; 1Þ and e3 ¼ ð1; 1Þ as shown in Fig. 2. Let us denote Srn ðDÞ the space of piecewise polynomial functions of degree n and class C r defined on D. A ðrÞ

Tang et al. proposed in Tang et al. (2014) a high-capacity reversible steganography (CRS) scheme. They improved the INP scheme by using two values C min and C max in the calculation of the predicted values I. These two values are computed as follows:

function f is said to be of class C r if the derivatives f 0; . . . ; f exist and are continuous. The interpolation problem in the space Srn ðDÞ consists of determining the interpolate operator:

C max ¼ maxfC ði;jÞ ; C ði;jþ1Þ ; C ðiþ1;jÞ ; C ðiþ1;jþ1Þ g;

I¼

X

ca Bð:  aÞ

a2Z2

C min ¼ minfC ði;jÞ ; C ði;jþ1Þ ; C ðiþ1;jÞ ; C ðiþ1;jþ1Þ g: The maximum value C max and the minimum value C min are used to compute the reference value AD as can be seen in the following equation:

which interpolates a given function f on the vertices of the triangulation D,

Iði; jÞ ¼ f ði; jÞ

ð6Þ

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

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An explicit computation of the equation above leads to:

8 y > > > > > x > > > > > > <1 þ y  x B2 ðx; yÞ ¼ 2  x > > >2  y > > > > >1 þ x y > > > : 0

where B is the B-spline in Srn ðDÞ of compactly support on D. This problem is equivalent to determine the coefficients fca g, see Unser (1999), De Boor et al. (1993), De Boor et al. (1983) and Chui (1988). In this section, our interest is the interpolation probS01 ðDÞ

lem in the space of bivariate linear spline function on the three directional mesh D. For this, we define firstly the box-spline which forms the basis of this space. The Haar bivariate box-spline is defined by the following equation:

B1 ðx; yÞ :¼ v½0;1Þ2 ðx; yÞ ¼

ifðx; yÞ 2 R2 n ½0; 1Þ2 ;

0

1

B1 ðx  t; y  tÞdt; 8ðx; yÞ 2 R2 :

X

if ðx; yÞ 2 E if ðx; yÞ 2 F otherwise:

f ðaÞB2 ð:h  a þ ð1; 1ÞÞ

ð9Þ

a2Dp;q

which interpolates a given function f defined on X at the points of the set V, i.e.

Ih ðv i;j Þ ¼ f ðv i;j Þ;

for i ¼ 0;    ; p and j ¼ 0;    ; q: 2

Moreover, the interpolation error verifies kIh  f k1 6 Kh where K is a constant and k:k1 is infinity norm (see De Boor et al. (1993)).

v

interior of its support ð0; 1Þ2 as shown in Fig. 3b. The bivariate linear box spline on the three directional mesh of the plane, called Courant Hat box spline, is defined by integrating B1 along the e3 direction. We obtain the hat form shown in Fig. 4b while the support is shown in Fig. 4a.

B2 ðx; yÞ :¼

Ih ¼

v

from which we see that B1 is a piecewise constant polynomial function; its support is shown in Fig. 3a. It is discontinuous around the boundary of its support and assumes the constant value 1 in the

Z

ð8Þ

if ðx; yÞ 2 D

þv

v

þv

Let X :¼ fxi;j ¼ i;j 2 iþ1;j ; i – pg; Y :¼ fyi;j ¼ i;j 2 i;jþ1 ; j – qg be the set of midpoints of horizontal and vertical edges respectively and

1 ifðx; yÞ 2 ½0; 1Þ2 0

if ðx; yÞ 2 C

The family fB2 ð:  aÞ; a 2 Z2 g forms a basis of the space S01 ðDÞ of piecewise linear polynomial on the three directional mesh D. Let h 2 Rþ ; p; q 2 N n f0; 1g, and let Dpq be the restriction of the uniform triangulation hD on a rectangular domain X ¼ ½0; ph  ½0; qh. Let V ¼ fv i;j ¼ ðih; jhÞ=i ¼ 0;    ; p=j ¼ 0;    ; qg be the set of ðp þ 1Þ  ðq þ 1Þ interpolation points in the rectangular domain X. Since the Courant Hat box spline B2 satisfies B2 ð1; 1Þ ¼ 1 and B2 ði; jÞ ¼ 0 for all other vertices of D, then we obtain an explicit formula of the interpolation operator Ih

Fig. 2. Three directional mesh D.

(

if ðx; yÞ 2 A if ðx; yÞ 2 B

ð7Þ

þv

let Z :¼ fzi;j ¼ i;j 2iþ1;jþ1 ; i – p and j – qgbe the set of the center of all squares in the uniform triangulation Dpq , as shown in Fig. 5. By using the Eqs. (8) and (9), we obtain for i ¼ 0;    ; p and j ¼ 0;    ; q:

8 f ðv Þþf ðv Þ > I ðx Þ ¼ i;j 2 iþ1;j > > h i;j > > > < f ðv Þþf ðv Þ Ih ðyi;j Þ ¼ i;j 2 i;jþ1 > > > > > > : f ðv Þþf ðv Þ Ih ðzi;j Þ ¼ i;j 2 iþ1;jþ1 :

ð10Þ

Fig. 3. (a): The support of B1 , (b): The Haar box spline B1 .

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

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A. Benhfid et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

Fig. 4. (a): The support of B2 , (b): The courant hat box spline B2 .

Fig. 5. Points and vertices pixels.

In our method, the cover image C represents the signal which we interpolate C ði;jÞ ¼ f ðv i;j Þ. By using the Eq. (8), the virtual predicted pixel Iði;jÞ is obtained by the following equation:

Iði;jÞ ¼

k 8 jðO ði;j1Þ þOði;jþ1Þ Þ > > 2 > > > > >
þO

if i ¼ 2p; j ¼ 2q þ 1

ð3Þ Si;j , they ð3Þ ð1Þ Si;j ¼ Si;j 

ð2Þ

are calculated as follows: Lk ði;jÞ

2

ð2Þ Si;j

¼

ð1Þ Si;j

þ 2Lk ði;jÞ

and

. The corresponding stego pixel Si;j is determined ð1Þ

ð2Þ

ð3Þ

as the closest one to Oi;j among Si;j ; Si;j and Si;j :

Þ

ði1;jÞ ðiþ1;jÞ if i ¼ 2p þ 1; j ¼ 2q 2 > > > > > j k > > : ðOði1;j1Þ þOðiþ1;jþ1Þ Þ otherwise: 2

ð11Þ

ðrÞ

Si;j ¼ Si;j

ðkÞ

with r ¼ argminðjOi;j  Si;j jÞ k¼1;2;3

Following the example shown in Fig. 6, the original pixel is O0;1 ¼ 56, we hide the message msg ¼ 00111 of size L1 ð0; 1Þ ¼ 5 bits into the interpolated pixel I0;1 ¼ 97 by substituting the 5 LSBs. ð1Þ

3.2. The dissimulation using OPAP Method

ð2Þ

ð3Þ

We obtain S0;1 ¼ 103; we find S0;1 ¼ S10;1 þ 25 ¼ 135 and S0;1 ¼ S10;1 

The objective of applying (OPAP) is to minimize the error between the original image O and the stego image S. To dissimulate data, we calculate the error of interpolation as in CRS which was defined in the Eq. (5). Let Lk ði; jÞ ¼ ½log 2 ðjEk ði; jÞjÞbe the number of successive bits of the secret message to be embedded by substitution into the interpolated pixel Ii;j obtained by the linear box-spline on the directional mesh. When a message of size Lk ði; jÞ bits is hidden by substitution into the interpolated pixel Ii;j , the message resides in the Lk ði; jÞ least significant bits of the obtained pixel ð1Þ

ð1Þ

2Lk ði;jÞ to Si;j ; we obtain two new pixels which we name Si;j and

ð1Þ

which we name Si;j . Thus, the idea is to modify the bits of Si;j that do not carry the message (weights above Lk ði; jÞ) in order to get close to the original pixel Oi;j . This can be achieved by adding

25 ¼ 71. Since 71 is the closest value to O0;1 among 135; 103 and ð3Þ

71, then the stego pixel S0;1 ¼ S0;1 ¼ 71.This procedure allows us to reduce the difference from 103  56 ¼ 47 to 71  56 ¼ 15. 3.3. Hiding message using MAE method The Message Adaptive Error is a method based on the error between the original image and the interpolation resultant image. The message is not substituted int the LSBs of the pixel Ii;j interpolated by the method described in the Section 3.1; but it is added to Ii;j as a decimal value Msg. In the extraction, Msg is the difference between the stego pixel Si;j and the interpolated pixel Ii;j . Since

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

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Fig. 6. Reversible scheme with OPAP method using numerical values.

the sign of this difference is irrelevant, then we have two possibilities, we can add Msg or subtract it from Ii;j :

Si;j ¼ Ii;j  Msg: ð1Þ

To choose the convenient sign, we calculate both Si;j ¼ Ii;j þ Msg and

ð2Þ Si;j

¼ Ii;j  Msg, then we choose the closest one to the original

pixel Oi;j :

Si;j ¼

8 < Sð1Þ ¼ Ii;j þ Msg i;j

if

: Sð2Þ i;j

otherwise:

¼ Ii;j  Msg

ð1Þ

ð2Þ

jOi;j  Si;j j < jOi;j  Si;j j

Fig. 7 shows the same example used in the previous section; the value of the original pixel is O0;1 ¼ 56. We suppose that we hide the 5 bits of the message vector ‘‘00111” in the pixel I0;1 ¼ 97 predicted by the interpolation method. The decimal value of these 5 bits is ð2Þ

ð1Þ

Msg ¼ 7. Then, we find S0;1 ¼ 97  7 ¼ 90 and S0;1 ¼ 97 þ 7 ¼ 104. ð1Þ

ð2Þ

The differences are jO0;1  S0;1 j ¼ 48 and jO0;1  S0;1 j ¼ 34. Since 34 < 48, we choose then S0;1 ¼

ð2Þ S0;1

4. Experiment results In this section, we use the standard images shown in Fig. 8. They are 512  512 8-bit gray-scale images which are usually used in tests. We apply the methods explained in the previous sections to achieve the test and comparison based on the imperceptibility and the capacity of embedding. The capacity of embedding refers to the maximal quantity of information that can be hidden without generating perceptible artefacts on the cover image. It is expressed by the number of bits that can be hidden in the image or the number of bits that can be hidden in one pixel (bit per pixel bpp). The PSNR informs about the imperceptibility of the steganographic scheme. When it is beyond 30 dB, the human eye can not distinguish between the cover and stego images. This metric is calculated as follows:

PSNR ¼ 10log10

2552 MSE

! dB;

where

¼ 90.

Fig. 7. Reversible scheme with MAE method using numerical values.

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

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Fig. 8. Six 512  512 grayscale testing images.

MSE ¼

512 X 512  1 X 2

512

2 C i;j  Si;j :

NCC ¼

i¼1 j¼1

We use also the Image Fidelity ðIFÞ to evaluate the imperceptibility. The IF has to be very close 1; because the difference 1  IF measures the ratio of the error energy to the image energy. It is calculated as follows: M X N X 

IF ¼ 1 

Si;j  C i;j

i¼1 j¼1 M X N X C 2i;j

2

hcs gg Q ¼ 4 2   c s : hc þ h2s g2c þ g2s

hcs ; hc hs

ð12Þ

The parameters gc and gs are the mean values of the cover and stego images respectively, and hc ; hs and hcs are given by the following expressions

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uM1 N1 uM1 N1 uXX uXX 2 2 hc ¼ t C i;j  gc ; hs ¼ t Si;j  gs ; i¼0 j¼0

: hcs ¼

M 1X N1 X

i¼0 j¼0

   C i;j  gc Si;j  gs :

i¼0 j¼0

i¼1 j¼1

To estimate the similarity of the cover and stego image, we use the metrics Normalized Correlation Coefficient ðNCCÞ and the Qindex. These two parameters take values between 1 and 1. The closest they are to 1, the more similar are the images. When they are close to 0, the images are said uncorrelated. When it is close to 1, they are said opposite.

To evaluate our work, we perform a comparison to the NMI, INP and CRS methods. Tables 1 and 2 show the values of the capacity and the metrics of the imperceptibility obtained by using the NMI (Jung and Yoo, 2009), INP (Lee and Huang, 2012), CRS (Tang et al., 2014) and the proposed methods with OPAP and MAE. We used the images Lena, Mandrill, Airplane, Peppers, Lake and Tiffany, they

Table 1 The comparison of capacity (bpp) and PSNR between NMI, INP, CRS and proposed method based on (OPAP) and (MAE). Image

Metric

NMI (Jung and Yoo, 2009)

INP (Lee and Huang, 2012)

CRS (Tang et al., 2014)

Proposed OPAP

MAE

Lena

PSNR Capacity

34.8457 0.75875

35.936 1.2698

38.2489 1.7617

40.3762 1.8109

40.2854 1.8109

Mandrill

PSNR Capacity

27.0762 1.6232

27.5971 2.3201

28.5853 2.8916

29.7128 2.9432

30.7378 2.9432

Airplane

PSNR Capacity

33.0183 0.68324

33.659 1.1232

38.2641 1.5856

41.6837 1.637

41.3954 1.637

Peppers

PSNR Capacity

34.377 0.74458

35.3741 1.2716

36.3833 1.782

38.7154 1.8373

38.6007 1.8373

Lake

PSNR Capacity

32.5011 1.0373

32.5902 1.6381

33.6581 2.168

35.6969 2.2234

36.3886 2.2234

Tiffany

PSNR Capacity

34.5023 0.63644

35.4419 1.0877

38.2357 1.5759

39.5246 1.6367

40.1976 1.6367

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

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A. Benhfid et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

Table 2 The comparison of the IF, NCC and Q-index between NMI, INP, CRS and proposed method based on (OPAP) and (MAE). Image

Metric

NMI (Jung and Yoo, 2009)

INP (Lee and Huang, 2012)

CRS (Tang et al., 2014) OPAP

MAE

Lena

IF NCC Q-index

0.99881 0.99545 0.99543

0.99908 0.99644 0.99644

0.99946 0.99796 0.99794

0.99967 0.99873 0.99873

0.99966 0.99869 0.99869

Mandrill

IF NCC Q-index

0.9931 0.96386 0.96371

0.99381 0.96803 0.96801

0.99504 0.9749 0.97484

0.99619 0.98053 0.9805

0.99699 0.9843 0.98428

Airplane

IF NCC Q-index

0.99881 0.99061 0.9906

0.99888 0.99117 0.99116

0.99953 0.99633 0.99633

0.99977 0.99821 0.99821

0.9998 0.9984 0.9984

Peppers

IF NCC Q-index

0.99767 0.99307 0.99305

0.99758 0.99281 0.9928

0.9981 0.99442 0.9944

0.99897 0.99696 0.99695

0.99916 0.9975 0.9975

Lake

IF NCC Q-index

0.99744 0.99408 0.99408

0.99728 0.99374 0.99373

0.99782 0.99504 0.99501

0.99867 0.99695 0.99694

0.99898 0.99764 0.99764

Tiffany

IF NCC Q-index

0.99942 0.98465 0.98464

0.99949 0.98672 0.9867

0.99972 0.99281 0.9928

0.99982 0.99527 0.99527

0.99985 0.99595 0.99595

have different degree of texture which is an important criterion in the interpolation based stegnography. Results in Tables 1 and 2 indicate that the proposed methods provide the larger capacity; and at the same time, they have the highest imperceptibility. We proposed to interpolate by the average of the neighbouring pixels horizontally, vertically and diagonally to minimize the error between the interpolated and the original pixels; this gives a better image quality. Furthermore, by using the OPAP or the MAE, the stego pixel is the closest possible to the original one. Hence, the obtained PSNR by the proposed methods are the highest. The values of IF, NCC and Q-index in Table 2 obtained by the proposed methods are closer to their optimal value 1 than the three other methods. For the capacity, we adopted the same methodology used in CRS to calculate the error of interpolation. However, what made the difference with CRS in the obtained capacity is the interpolation technique we utilized; it allowed us to surpass the capacity of CRS; because the interpolation in CRS is concentrated on the minimum pixel of the four reference pixels; whereas in the proposed method, the interpolation is the average in the three directions. Fig. 9 shows the evolution of the PSNR as we increase the size of the secret message from 0:1 bpp to 1 bpp for the six test images shown before in Fig. 8. As can be seen in Fig. 9, the proposed work using OPAP and MAE offers better trade-off between the PSNR and the capacity of hiding. To evaluate the security of the proposed steganographic scheme, we use the statistical attack v2 (Westfeld and Pfitzmann, 1999). It is based on the idea that at the end of the dissimulation process, the sum of the frequencies of each pair of pixels (2i; 2i þ 1) in the histogram remains unchanged; and the frequency of 2i and 2i þ 1 tends to the average of their initial values. Let m2i and m2iþ1 be the frequencies of 2i and 2i þ 1 in the histogram m þm respectively. Their average is gi ¼ 2i 2 2iþ1 . The v2n statistic with n degree of freedom is then calculated as follows:

v2n ¼

nþ1 X ðm2i  g Þ2 i¼0

gi

i

:

The degree of freedom is the number of pairs ð2i; 2i þ 1Þ having a sum of frequencies greater than 4, i.e.

n ¼ cardfð2i; 2i þ 1Þ; 0 6 i 6 127; m2i þ m2iþ1 > 4g:

Proposed

Then, the probability that the attacked image hides a secret data is computed as follows:

p¼1

n 2

1  n

2C

2

Z v2n

t

n

e2 t21 dt;

0

while C is the Gamma function defined for every positive x by

CðxÞ ¼

Z

þ1

tx1 et dt:

0

Table 3 shows the PSNR and v2 values of the same six images for the proposed methods. To measure the contribution of the OPAP and MAE, we use only the LSB substitution which is the simplest way of insertion; we hide data by substituting the LSBs of the interpolated pixels. Results are shown in the column LSB for both PSNR and v2 test. Obviously, the PSNR of the LSB substitution is the lowest. This shows the effectiveness of the proposed MAE and OPAP; they search for the closest pixel possible to the original pixel that can hide data. We obtain an amelioration of the PSNR with 3 dB. The v2 results in Table 3 indicate a very low rate of detectability; the proposed work is not detected by this attack. The changes of the histogram are not predictable by v2 . In Table 4, the proposed work is compared to schemes developed in Ou et al. (2012), Pei et al. (2013), Wang et al. (2013) and Hu and Li (2015). The comparison is done on the images Lena, Mandrill and Airplane. As can be seen, the proposed methods OPAP and MAE outperform the other methods in the PSNR and the capacity at the same time. The proposed work surpasses the work developed by Ou et al. (2012) in both PSNR and the capacity for the images Lena and Airplane. The difference in PSNR is 5 and 7 dB respectively, the capacity we obtain is respectively 1:6 times and 1:3 times greater. For the image Mandrill, while the PSNR of the proposed work is smaller by 9 dB, the capacity is almost 6 times greater. The PSNR obtained by Wang et al. (2013) is constant for the three images with a significant variation of the capacity of hiding. It is higher than our PSNR with 8 dB but our capacity is more than 6 times greater than theirs. In comparison to Pei et al. (2013), we obtained a comparable capacity for Lena and Airplane but a remarkably greater PSNR. For Mandrill, the proposed work gives better PSNR

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

9

A. Benhfid et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

Fig. 9. Comparison of all interpolation methods with the two proposed methods over six standard testing images.

Table 3

v2 results on six usual images after using OPAP and MAE. Metric

v2 (Westfeld and Pfitzmann, 1999)

PSNR

Image

LSB

MAE

OPAP

LSB

MAE

OPAP

Lena Mandrill Airplane Peppers Lake Tiiffany

37.2916 27.2397 38.6339 35.5388 32.7401 36.7134

40.2854 30.7378 41.3954 38.6007 36.3886 40.1976

40.3762 29.7128 41.6837 38.7154 35.6969 39.5246

4.1481e10 0.5239 0 6.0263e13 4.8950e13 0

1.3890e08 0.8667 0 1.9386e11 0 0

6.5503e15 2.2922e04 0 0 0 0

and remarkably better capacity. In Hu and Li (2015), while the capacity increases with only 0:2 bpp from Lena and Airplane to Mandrill, the PSNR decreases with 4 dB. The capacity provided by the proposed work increases significantly for images with a lot of texture as can be seen in comparison between Lena and Mandrill,

the capacity grows by 61%. Results in this table indicate that the proposed work gives better compromise between the PSNR and the capacity of hiding than the other works. We obtain around 40 dB in PSNR for almost 2 bpp capacity and 30 dB for almost 3 bpp.

Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016

10

A. Benhfid et al. / Journal of King Saud University – Computer and Information Sciences xxx (2018) xxx–xxx

Table 4 Comparison of our proposed work with four reversible data hiding methods basing on image quality and capacity. Methods

Metric

Lena

Mandrill

Airplane

Ou et al. (2012)

PSNR Capacity (bpp)

35.3729 1.0954

38.9982 0.5321

34.1648 1.2242

Wang et al. (2013)

PSNR Capacity (bpp)

48.6747 0.2716

48.9441 0.0952

48.7547 0.3332

Pei et al. (2013)

PSNR Capacity (bpp)

26.8483 1.8388

21.4183 1.2496

26.7403 1.8108

Hu and Li (2015)

PSNR Capacity (bpp)

34.6967 1.6945

30.1452 1.8483

34.2991 1.6636

Proposed (OPAP)

PSNR Capacity (bpp)

40.3762 1.8109

29.7128 2.9432

41.6837 1.637

Proposed (MAE)

PSNR Capacity (bpp)

40.2854 1.8109

30.7378 2.9432

41.3954 1.637

5. Conclusion In this paper, we introduced new data hiding methods that allowed us to increase the capacity while keeping the imperceptibility at a good level. Depending on the image’s complexity, theadopted interpolation technique and the method utilized to calculate the interpolation error enabled us to obtain a good capacity. Results show that the capacity and PSNR of the proposed methods whether using OPAP or MAE are higher than NMI, INP and CRS algorithms. In addition, the v2 test results prove the undetectability of the proposed algorithm. Therefore, these substantial performance improvements demonstrate the effectiveness of the proposed scheme. In comparison to wavelet transform based steganographic schemes, the wavelets forbid to dissimulate data in the bloc of the approximation; this subtracts quarter of the image size from the capacity. However, the imperceptibility is good. The proposed work has better capacity but lower imperceptibility than wavelet based schemes. In our future works, we will focus on strengthening the robustness of our work and study its resistance to attacks trying to destroy the information hidden inside the stego image.

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Please cite this article in press as: Benhfid, A., et al. Reversible steganographic method based on interpolation by bivariate linear box-spline on the three directional mesh. Journal of King Saud University – Computer and Information Sciences (2018), https://doi.org/10.1016/j.jksuci.2018.09.016